#### 3.1. Sensitivity to Averaging Methods and Averaging Intervals

Annual SIE maximums and minimums are commonly used sea ice climate indicators that can be derived from sea ice concentration products. The sensitivity of Arctic SIE trends to averaging methods and intervals is demonstrated by performing linear trend analysis on the time series of averaged annual SIE maximums and minimums. The linear trend line of annual Arctic SIE maximums and minimums for the case of the five-day moving average time series is very close to that of the original daily time series. The subset-averaged maximum/minimum values tend to be slightly reduced/increased as compared to those from moving-averaged (

Figure 1). Furthermore, the decadal trends from subsetting averages exhibit slightly higher variability as compared to those from moving averages (

Table 1).

The percentage changes of the linear decadal trends relative to the trends of the original daily time series tend to be more variable for the SIE annual maximums than the SIE annual minimums. The linear decadal trends of the SIE annual maximum from 30-day moving and subsetting averages are 4.05% and 6.07% smaller, respectively, relative to that from the original time series (

Table 1). This means that applying either averaging technique results in a more conservative sea ice loss trend. The percentage change of the linear decadal trends from of the original time series for SIE annual minimum is very small (less than 0.92%;

Table 1).

The decadal trend lines from 10-day averages are very close to those of the five-day averages while slight differences are present for those from 30-day averages (

Figure 2). Note that the seven-day average case is not included in

Figure 2, as its trend lines are between that of the five-day and 10-day averages; however, its statistical attributes are still captured in

Table 1.

The percentage changes of the SIE annual maximum decadal trends relative to those from the original daily SIE time series range from about −2.02% to 4.05% for the moving average, and from −0.58% to 6.07% for the subsetting average (

Table 1). The percentage changes are much smaller for the SIE annual minimum decadal trends. They range from −0.12% to 0.81% for the moving averages and from −0.92% to 0.12% for the subsetting averages. The same order of sensitivity is found for the decadal trends from the times series from the 10-day and 30-day averaging intervals relative to those of the five-day averaging interval (

Figure 2). Relative to the moving averages with the same averaging interval, the percentage changes for the subsetting averages range from −2.29% to –0.29% for the SIE annual maximums, and from 0.12% to 0.81% for annual minimums (

Figure 1). Therefore, the overall sensitivity is 6.61%/1.04% or less for the decadal trends of the SIE annual maximums/minimums, respectively.

#### 3.2. Sensitivity of Arctic Ice-Free Projection to Time Domain of Linear Regression

As shown in the previous section, no significant difference amongst trends calculated from 5-, 7-, 10-day and 30-day moving and subsetting averages was found. Since the five-day moving average is a commonly used approach, we will use the SIE annual minimum time series derived from the five-day moving average method for the rest of the analysis. The linear regression trends for different data periods are computed, from which the FIASY is projected. The results are displayed in

Figure 3 and summarized in

Table 2.

The linear trend for the whole data period (1979–2015) is about −0.87 (10

^{6} km

^{2}/decade) (

Table 2 and

Figure 3). Assuming this trend is persistent, the Arctic summer is projected to be ice-free after 2058 (zero-crossing at year 2069). The linear trend from 1996 to 2015 (the last 20 years of the available time series) is about −1.47 (10

^{6} km

^{2}/decade), which projects the Arctic summer to be ice-free after 2036, in less than 20 years (zero-crossing at year 2043). In comparison, the trend for the current climate normal period (1981–2010) is −0.82 (10

^{6} km

^{2}/decade), which would result in Arctic ice-free summer beginning at year 2062 (zero-crossing at 2074). The trend of the climate normal period is slightly slower than that based on the trend from the whole record period (1979–2015) as the time period for the climate normal does not include the record low in 2012. On the other hand, the linear trend from the first 20 years of the time series (1979–1998) projects 2147 for ice-free and 2174 for zero-crossing. Therefore, it seems that the accelerated reduction of sea ice extent in the last 20 years (1996–2015) has put the prospect of an ice-free Arctic summer well into our near future. Although all of the above linear trends for the annual SIE minimums are significant at the 99% confidence level, the question is whether any of these linear trends will persist in the future. There is a drastic jump in decadal trends between the first 20 and last 20 years (−0.38 vs. −1.47 10

^{6} km

^{2}/decade). The acceleration, even in the linear sense, is very distinct and is reflected in the ice-free projections shown in

Figure 3. If this trend-acceleration continues, the FIASY could continue to move closer to the present day.

#### 3.3. Sensitivity of Different Statistical Curve Fitting Functions

The linear fitting is the simplest approach. However, visual examination of the SIE annual minimum data presented in

Figure 3 suggests that the linear functional form is not appropriate for this application, as the decay in SIE is clearly accelerating in the latter part of the time series.

Statistical curve-fitting models for Arctic sea ice extent and their associated FIASY projections have been previously examined by several studies (e.g., [

10,

14,

15]). In this study, using the same long-term, consistent time series of sea ice data, we compare six commonly used statistical models for this type of application and present the results in

Figure 4 and

Table 3.

Metrics from model optimization are captured in

Table 3, including RMSE (in) (root mean squared errors over data within the time period); RMSE (out) (root mean squared errors over data outside of the training domain (e.g., for the 1979–1998 period, this would be calculated for 1999–2015)); AICc (Akaike information criterion corrected for small sample size); and W-Akaike weights [

19,

20]. The AICc is a statistically-based means of model selection that estimates the quality of each model in comparison to other models for a given data set, taking into account the number of model parameters and data set sample size. Given a set of AICc values, the optimal model of the sample has the minimum value. W-Akaike weights are derived directly from the AICc values and may be interpreted as the probability that the model is the best of the sample. Note that summing the W-Akaike weights across the sample is 1, and further that the optimal model of the sample has the maximum (probability) value.

Table 4 and

Table 5 summarize the ice-free and zero-crossing projections, respectively, for each of the models as optimized over the 6 different time periods. Note that the Gompertz model will never cross zero, denoted “N/A” in the

Table 5 entries. Additionally, when tuned over the first 20 years and the last 20 years, the quadratic model also failed to reach both ice-free and zero-crossing conditions. These cases indicate that, although the model fit data well over the training period, future extrapolation is likely not reliable. Ice-free conditions are projected to occur anywhere from 2014 to beyond 2100. Similarly, zero-crossing is projected to occur anywhere between 2015 to beyond 2100.

Examination of

Figure 4a clearly shows that although all are about equally good at representing the period of calibration, none of the models trained over the first 20 years of the time series do a good job of predicting future SIE annual minimums. Mostly they tend to overpredict the sea ice extent; the quadratic model actually projects that SIE will increase over time, which is unlikely given that the data for the final 17 years clearly show a decreasing trend. In

Figure 4b, the exponential, Gompertz, quadratic, and linear with lag models all appear to characterize well the domain of calibration, 1979–2008. However, comparing the extrapolated values of 2009–2015 with model predictions shows that none of these four candidate models capture the future behavior well. In particular, the exponential model drops off far too sharply, yielding a prediction of ice-free conditions in 2014, which we know did not occur. These poor predictions are quantitatively reflected in the RMSE (out) value shown in

Table 4, which is significantly higher than for the other candidates. Therefore, although this model was selected as best among the candidates for this calibration domain, little faith is held in its ability to project future Arctic sea ice conditions.

The model fits resulting from fitting over the climate normal period of 1981–2010 are shown in

Figure 4c. Only two data points not used for calibration are available for comparison in advance of the calibration domain and five data points after it. The quadratic and linear with lag models appear unstable extrapolating into the past. The exponential and Gompertz models appear to underestimate the sea ice extent in the future. The log and linear models do not appear to capture the dynamics of any of the curve that well, which is corroborated by their low W values in

Table 3.

In

Figure 4d, the entire available time domain is used for calibration, so no inferences may be made about the models’ prediction capabilities. The three best fitting models are the quadratic, Gompertz, and exponential, respectively. However, the quadratic model is known to be unstable in extrapolation regions and given that there is no reserved data to demonstrate otherwise, we assume it cannot be fully trusted to provide reliable predictions.

Using the last 30 years of the time series, 1986–2015, yields models predictions from all candidates that look quite similar during the calibration domain (

Figure 4e). It also yields the best overall predictions compared to other periods, reflected by the smaller RMSE values (

Table 3). It is therefore not surprising that the linear model, the simplest of all the six candidates, is selected as best using the statistical metrics that take into account and penalize models with greater complexity and the associated larger number of estimated parameters. However, the linear model performs the worst of all candidates in extrapolation as evidenced by the RMSE (out) value in

Table 3. Admittedly, this metric only accounts for comparisons taking place in the past (1979–1985) and our more pressing concerns are with the model’s ability to extrapolate into the future. Fitting over the last 20 years, 1996–2015, yields varying results. None of the models fit data not used for calibration well (

Figure 4f and

Table 3). The quadratic model is particularly unstable for extrapolation in either direction, in part demonstrated by the large RMSE (out) value in

Table 3. Therefore, although the associated W value implies it is the most probable candidate for 1996–2015, it cannot be trusted outside of this period.